ivreghdfe command for system GMM

[Frank Giaquinto]

I am attempting to estimate a persistent panel data model using the System GMM estimator, as proposed by Bundell and Bond in 1998. Consequently, the regression comprises the lagged dependent variable as a predictor, alongside other predictors.

Is it possible to estimate heterogeneous coefficients for my predictors, including the coefficients for the lagged dependent variables? This would allow for heterogeneous slopes, as discussed by Correia in 2016, who introduced an estimator for linear models with multi-way fixed effects.

Would the ivreghdfe command be suitable for this purpose?

[Sebastian Kripfganz]

If you create the relevant instruments as new variables first, this might work. With my xtdpdgmm command, you can estimate the model by system GMM with homogeneous slopes. Subsequently, the postestimation command predict with option iv allows you to obtain the instruments as new variables, which can then be fed into ivreghdfe; see slides 39 and following in my 2019 London Stata Conference presentation:

• Kripfganz, S. (2019). Generalized method of moments estimation of linear dynamic panel data models. Proceedings of the 2019 London Stata Conference.

[Frank Giaquinto]

Thank you for your valuable insights. I've carefully read your slides and I am currently working on estimating heterogeneous slopes using your provided System-GMM syntax on page 38.

xtdpdgmm L(0/1).n w k, model(diff) collapse gmm(n, lag(2 4)) gmm(w k, lag(1 3)) gmm(n, lag(1 1) diff model(level)) gmm(w k, lag(0 0) diff mode l(level)) two vce(r)

The Stata syntax you provided appears to be specifying a dynamic panel data model. Allow me to explain my understanding (I'm sorry for the naive question, but I've just recently moved to STATA, I've always used R). The model you are estimating is:

\begin{equation}
n_{it} = \rho n_{i,t-1} + \beta_1 w_{it} + \beta_2 k_{it} + \alpha_i + \delta_t + \nu_{it}
\end{equation}

with individual specific unobserved heterogeneity and time fixed-effects.
Then, you perform an estimation using a system-GMM approach, where equations are specified in both first-differenced form and levels.
For the first-differenced equations, lags from 2 to 4 of n are utilized as instruments for Δ n_i,t-1. Additionally, you assume that w and k are predetermined regressors, and their first differences are instrumented by the corresponding values at levels from lag 1 to 3.
Assuming stationarity in the first moments of the series y,w,k, you utilize (T-2) + 2(T-1) additional non-redundant moment restrictions for the equations in levels: Δy_i,t-1 Δw_it, and Δ k_it are the respective instruments.
The standard errors subsequently follow the Windmeijer correction (2005). Are my statements correct?

Then, with the command

quietly predict iv*, iv

I can extract the instruments used in the system GMM estimation.

Suppose I aim to introduce heterogeneity in the slope coefficient of the lagged value of n across individual IDs. However, I am uncertain about how to provide the ivreghdfe function with this particular set of instruments.According to the Stata help page, ivreghdfe is essentially ivreg2 with an additional absorb(). How can I differentiate between the instruments intended for the first-differenced equations and those designated for the equations in levels?

[Jeff Wooldridge]

Allowing heterogeneity in the slopes when you have a lagged dependent variable is tricky. Method of moments solutions generally do not produce consistent estimators because the slope must more correlated with y(i,t-1). (If you have a large T, then you can estimate separate slopes for each i, but then you don't need to do IV in the first place.) There are maximum likelihood solutions, but I think they'd have to be programmed. You'd have to model the distribution of a(i),b(i)] -- the intercept and slope -- conditional on the initial conditions and the strictly exogenous variables. A general discussion is in my 2005 Journal of Applied Econometrics paper.

You might start with something easier, such as interacting y(i,t-1) and observed heterogeneity -- that is, exogenous variables -- and see if anything shows up.

[Frank Giaquinto]

Dear Professor Wooldridge,
I appreciate your insights. I was considering exploring a somewhat simpler approach, but I am uncertain about its correctness, and I sincerely apologize if my inquiry seems naive.

Suppose I posit that the true Data Generating Process (DGP) is articulated as follows:
\begin{equation}
\Delta y_{it} = (\rho + \gamma_2 R2_j + \ldots + \gamma_Z RZ_j)y_{i,t-1} + x_{it} \beta + w_{it} \phi + \alpha_i + \delta_t + \nu_{it}
\end{equation}
where Rs_j =1 if s=j and zero otherwise. Essentially, j=1,2,…,Z , with Z<N. In other words, if i denotes the country and j represents the region, I am allowing for heterogeneity in slopes across regions.
Furthermore, x is a vector of preditermined regressors and w is a vector of strictly exogenous regressors.

I am eager to hear your expert opinion on whether such a specification makes sense. Additionally, do you believe it is feasible to estimate this dynamic panel through a system-GMM approach?

[Frank Giaquinto]

Dear Professors,
any suggestion?
Thanks a lot

[Jeff Wooldridge]

That's essentially trying to apply fixed effects to a model with lagged dependent variable -- but in this case, you're treating the coefficients on lagged y as parameters. This will not have good statistical properties in general. If T is large enough, then it might.

[Sebastian Kripfganz]

If the number of countries i within each region j is reasonable large, then you might still be able to do this with relatively small T. As Jeff mentioned, the easiest way to implement this would probably be to simply interact the lagged dependent variable with the region dummies.

Your understanding of the xtdpdgmm command syntax is correct. If you want to explicitly include time dummies (when T is not too large), you can specify the teffects option.

Regarding the instruments generated with the predict command, notice that the command does not actually create instruments for the first-differenced model, but instead it transforms these instruments back into suitable instruments for the level model. That is, all variables created by the predict command should be treated as instruments for the level model; some of them (those specified initially for the first-differenced model) will be orthogonal to any time-invariant variables. I hope this makes sense; see also slide 33 of my presentation.

[Jeff Wooldridge]

Agree with Sebastian. I read too quickly and didn't see you were using countries nested within regions. As Sebastian says, you can make this work if you have enough countries per region.

[Frank Giaquinto]

Dear professors,
I would like to express my sincere gratitude for your valuable responses. I have a final inquiry regarding the interaction of the regional dummy with the lagged dependent variable. Specifically, when introducing the variable R2₂ y_i,t-1 (representing region 2 dummy interacted with y lagged by one period), it is evident that this variable is endogenous and lagged levels of order 2 and higher, as well as lagged first differences of yi,t-1 shall be used as instruments for R2₂ y_i,t-1 for the equations in the first differences and levels, respectively.
Thus, would the notation c.L.y#c.R2 be correct for adding interaction terms into the xtdpdgmm command?
Thanks

[Sebastian Kripfganz]

Sounds alright to me.

[Frank Giaquinto]

Thanks a lot!